C o n t r a c t No. W-7405-eng-26

R e a c t o r D i v i s i o n

SOLUTION OF THE EQUATION DESCRIBING THE INTERFACE B E T " TWO

FLUIDS FOR THE VOLUME AND PRESSURE WITHIN ATTACHED,

SESSILE SHAPED, BUBBLES AND DROPS

3. W. Cooke

This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the United States Atomic Energy Commission, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, com- pleteness or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe ptivately owned rights.

AUGUST 1971

OAK RIDGE NATIONAL LABORATORY Oak R i d g e , Tennessee

O p e r a t e d by UNION CARBIDE CORPORATION

f o r t h e U. S. ATOMIC ENERGY COMMISSION

V

iii

CONTENTS

Page

Abs t r ac t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

In t roduc t ion . . . . . . . . . . . . . . . . . Deriva t ion of t h e I n t e r f a c i a l Equation . . . . Numerical So lu t ion of t h e I n t e r f a c i a l Equation

Computer Program . . . . . . . . . . . . . . . Solu t ion of t h e D i f f e r e n t i a l Equation . . Solu t ion f o r h/a and V/a3 versus fl and qj . Solu t ion f o r h/a and V/a3 versus fl and r/a

So lu t ion for h/a and V/a3 versus r/a . . . Range. Running Time. and Accuracy . . . .

. . . . . . . . . . . 1

. . . . . . . . . . . 2

. . . . . . . . . . . 5

. . . . . . . . . . . 7

. . . . . . . . . . . 7

. . . . . . . . . . . 7

. . . . . . . . . . . a

. . . . . . . . . . . 8

. . . . . . . . . . . 10

Resu l t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . 17

Estimation of t h e Accuracy . . . . . . . . . . . . . . . . . . . 17 Comparison with Previous Resul t s . . . . . . . . . . . . . . . . 1 7

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

A . Parametr ic Crossp lo ts . . . . . . . . . . . . . . . . . . . 27 B . Tabulated Resul t s . . . . . . . . . . . . . . . . . . . . . 33

C . Computer Program . . . . . . . . . . . . . . . . . . . . . . 43

SOLUTION OF THE EQUATION DESCRIBING THE INTERFACE BETWEEN TWO

FLUIDS FOR THE VOLUME AND PRESSURE WITHIN ATTACHED,

SESSILE SHAPED, BUBBLES AND DROPS

J. W. Cooke

ABSTRACT

A numerical computer program w a s w r i t t e n t o so lve t h e equa- t i o n desc r ib ing t h e i n t e r f a c e between two immiscible f l u i d s t o o b t a i n t h e shape, s i z e , volume, and pressure of a t t ached bubbles and d r o p l e t s . These r e l a t i o n s h i p s a r e important t o t h e s tudy of three-phase hea t t r a n s f e r , superheat , c r i t i c a l cons tan ts , and i n t e r f a c i a l energ ies . Previous s o l u t i o n s have been obtained wi th l imi t ed accuracy f o r a r e s t r i c t e d number and range of v a r i - ables. The p resen t resul . ts a r e given i n bo th g raph ica l and t a b u l a r form f o r a wide range and number of parameters, and t h e computer program is included so t h a t an even broader range and number of v a r i a b l e s , as wel l as s p e c i f i c va lues , can be obtained as requi red . I n p a r t i c u l a r , a n express ion f o r t h e maximum- bubble-pressure was derived which is considerably more accu ra t e over a wider range than previous express ions .

Keywords. Surface tens ion , i n t e r f a c i a l t ens ion , bubbles , drops, contac t angle , maximum-bubble-pressure, f lu id-vapor in - t e r f a c e .

INTRODUCTION

A knowledge of r e l a t i o n s h i p s among volume, pressure , shape, and s i z e

of a t t ached bubbles and d r o p l e t s are important i n t h e s tudy of b o i l i n g

and condensing hea t t r a n s f e r , superhea t and c r i t i c a l po in t phenomena,

mass t r a n s f e r s t u d i e s , and i n t h e measurement of con tac t angles and sur-

f a c e t e n ~ i 0 n . l ~ ~

of t h e second-order, nonl inear d i f f e r e n t i a l equat ion desc r ib ing t h e i n t e r -

f a c e between two f l u i d s .

a n a l y s i s f o r very small bubble s i z e s by Rayleigh5 and SchrGdinger,' by

numerical hand c a l c u l a t i o n s f o r a wide range of d i s c r e t e drop shapes by

Bashforth and Adam,7 and by numerical computer c a l c u l a t i o n s f o r d rop le t

volumes and he igh t s by Baumeister and H a m i l l . 8

SchrGdinger s o l u t i o n is no longer used ex tens ive ly because of concern

regard ing i t s accuracy; and t h e Bashforth and Adam (as w e l l as t h e

These r e l a t i o n s h i p s can be obta ined from t h e s o l u t i o n

This equat ion has been solved by pe r tu rba t ion

However, t h e Rayleigh-

1

2

Baumeister and H a m i l l ) s o l u t i o n s were obtained f o r a l i m i t e d number

range of variables.

I n order t o check t h e accuracy and t o extend t h e use fu lness O f

and

t h e

above so lu t ions , a numerical computer program w a s w r i t t e n t o so lve t h e

i n t e r f a c i a l equat ion . The s u b j e c t program so lves t h e i n t e r f a c i a l equat ion

f o r any value of t h e dimensionless shape parameter, p, f o r p o s i t i v e va lues

of t h e te rm @Z ( s e s s i l e drops or above-attached bubbles)

previous s o l u t i o n s f o r @ > 180 degrees .

t h e dimensionless X, Z, @ coord ina tes desc r ib ing t h e p r o f i l e s of t h e i n t e r -

face, b u t a l s o gives t h e bubble volume w i t h i n and t h e pressure d i f f e r e n c e

ac ross t h e i n t e r f a c e as a func t ion of p r o f i l e shape and, i n add i t ion , t h e

maximum pressure d i f f e rence ac ross t h e i n t e r f a c e f o r a given r ad ius of

attachment (which is requi red f o r t h e c a l c u l a t i o n of t h e su r face t ens ion

by t h e maxim-bubble-pressure technique) .

* and extends t h e

The s o l u t i o n not only provides

T h i s r e p o r t descr ibes t h e d e r i v a t i o n of t h e i n t e r f a c i a l equat ion,

i t s numerical s o l u t i o n , and t h e computer program employed.

s u l t s and a n e s t ima t ion of t h e i r accuracy a r e presented and compared

wi th t h e previous s o l u t i o n s .

The r e -

DEWXATION OF THE INTERFACIAL EQUATION

A t t h e i n t e r f a c e sepa ra t ing two immiscible f l u i d s (usua l ly a l i q u i d

arid i t s vapor) , a fo rce imbalance e x i s t s which r e s u l t s i n a n i n t e r f a c i a l

t ens ion .

d i r e c t i o n s tangent t o t h e i n t e r f a c e . Neglect ing g r a v i t a t i o n a l fo rces , a

f o r c e balance on a n i n f i n i t e s i m a l segment of a f r e e su r face i s shown i n

F ig . 1-a.

where rl and r2 a r e t h e p r i n c i p a l r a d i i of curva ture .

would thus be:

The t e n s i o n is a fo rce p e r u n i t l eng th p u l l i n g uniformly i n a l l

The su r face i s bowed by a uniform d i f f e r e n t i a l p ressure , P, The fo rce balance

P & & = 2 y Ax-- + & ( *

The case f o r negat ive values of t h e below-attached bubbles) w i l l be presented

”) 2r2 ,

term gZ (pendant drops or i n a l a t e r r e p o r t .

3

ORNL- DWG 7f - 7609

f I I \

PAZAX

\ \ I I

( b ) BUBBLE ’ Fig. 1. Schematic View of a Sessile-Type Drop and Bubble

and the Force Balance on an Infinitesimal Surface Element.

4

o r

P = ( k + k ) .

If g r a v i t a t i o n a l fo rces a r e present , t h e inf luence of t h e d i f -

f e r e n t i a l f l u i d dens i ty , p, must be considered. For a n i n t e r f a c e assumed

t o be symmetrical about an a x i s of revolu t ion , t h e fo rce balance equat ion

f o r t he bubble shown i n Fig. 1-b w i l l be

PQ P - .( i + k) - - ( A - Z ) = O

gC

A t t h e orgin, rl = r a E b and thus 4, - where p =

2Y Pg

gC

P = - + A - b

Furthermore, wi th rl = x / s i n GY Eq. (3) becomes

- a + - - y ( T + - - ) = o Pg= s i n @ 1 . b gC

(3)

(4)

(5 )

A similar d e r i v a t i o n f o r t h e drop shown i n F i g . 1-b would a l s o r e s u l t

i n E q . ( 5 ) . b, and a f t e r rearranging:

This equat ion can now be made dimensionless wi th r e spec t t o

1 s i n C$ - + - = 2 + R X

where

and

Equation (6 ) can be t ransposed i n t o

s u b s t i t u t i n g

Z

and Z = - . b

Car t e s i an coord ina tes by

5

1 d2 Z/dx“

and

R [l + (dZ/dX)”IY”

dZ/dX s i n 9 5

2 1 / a c 1 + ( d Z / W 1

t o o b t a i n

= (2 + pz) [l +(3IV2 , ( 8 ) d2 Z

which i s a second-order nonl inear d i f f e r e n t i a l equat ion, w i t h boundary

condi t ions :

x = o ; z = o X = 0 ; dZ/XdX = 1

Although Eq. (8) cannot be solved a n a l y t i c a l l y i n terms of ordinary

func t ions , i t s numerical s o l u t i o n is descr ibed i n t h e next s e c t i o n .

NUMERICAL SOLUTION OF THE INTERFACIAL EQUATION

Equation (6) can be rearranged t o read

1 A. -

2 + pz - ( s i n @)/x

and by d e f i n i t i o n :

dx - = R COS @ = F (X, Z, 6) d9

,

dZ - R s i n 9 = G (x, Z, $) dd

(9)

A s long as gZ > 0, R w i l l always be f i n i t e .

6

A numerical technique of fourth-order accuracy developed by Runge-

Kutta’ wu:; s e l e c t e d t o zolve the s e t of simultaneous Eqs . (9), (lo), and (11) ; the i t e r a t i v e equat ions decr ib ing t h i s technique a r e :

1

G + 2kl + 2k2 + k3) + O ( b $ > ” , ‘n+l = ‘n + - (ko

1 ~

= z ’n+l n 6 + - (mo + 2ml + 2m2 + Q ) + o ( w ) ~ ,

where

1 1 1

1 1 1

- 1 1 1

and where t h e symbol O ( A $ ) ~ r ep resen t s a term which i s small, of t h e

order (!!4)5, when A$ i s smll.

Equations (12) and (13) a r e of a form t h a t can be r e a d i l y t r a n s -

formed i n t o a conputer program.

7

C O M P U T E 3 PROGRAM

The computer program f o r t h e s o l u t i o n of t he i n t e r f a c i a l equat ion

and f o r t h e c a l c u l a t i o n of t h e var ious output parameters i s l i s t e d i n

Appendix C .

below.

The program cons is ted of f o u r p a r t s which a r e d iscussed

So lu t ion of t h e D i f f e r e n t i a l Equation. Two subrout ines i n both

s i n g l e and double p r e c i s i o n and cons i s t ing of four i t e r a t i o n loops were

w r i t t e n t o so lve Eqs . (12) and (13) . The subrout ines RHOS and RHOD

c a l c u l a t e t h e values of R and supply t h e values of t h e func t ions F and

G t o t h e subrout ines RUNGKS and RUNGKD. These l a t t e r subrout ines c a l -

c u l a t e t h e values of t h e c o e f f i c i e n t s k. and m. and t h e new va lues of

X, Z, and @ f o r r e in t roduc t ion i n RHOS and RHOD t o cont inue t h e i t e r a t i v e procedure. The i t e r a t i o n procedure i s i n i t i a t e d by equat ing

X, Z, and @ t o zero and R t o one.

1 1

Solu t ion f o r h/a and V/a3 versus p and @. The p res su re and volume

w i t h i n t h e a t t ached bubble a r e ca l cu la t ed from X, Z, and @. The pressure

r e l a t i o n i s given by Eq. (4) and can be s impl i f i ed by us ing Eq. (7) and

t h e d e f i n i t i o n of t h e s p e c i f i c cohesion,

t o ob ta in

h/a =Jq + z r & p , where

and Z = Z ( x = r ) = A , r

h = gcP/Pg

The volume r e l a t i o n can be obtained from t h e i n t e g r a t i o n of

d(E3) = IT X2 dZ ,

(1-5)

where X and dZ can be obtained from Eq. (6 ) . equat ion by p a r t s gives t h e r e l a t i o n s h i p :

In te 'g ra t ing t h e r e s u l t i n g

8

v & 2 s i n $I

b" @ X

Upon s u b s t i t u t i n g E q s . ( 6 ) , (14), and (15) i n t o

express ion f o r t h e dimesionless volume i s obtained:

where

is t h e dimensionless r ad ius of a t tachment .

So lu t ion f o r h/a and V/a3 versus @ and r/a. To o b t a i n the pres-

r u r e and volume wi th in a t t ached bubbles f o r a given r ad ius of a t tachment

fyom t h e numerical s o l u t i o n s X, Z, @, of t h e i n t e r f a c i a l equat ion,

s e x r a l cond i t iona l "IF" statements were requi red . The i n t e r r e l a t i o n -

s k i p of h/a, 9, r/a, and @ a r e shown schemat ica l ly i n F ig . 2.

i t e r a t i v e s o l u t i o n of t h e i n t e r f a c i a l equat ion proceeds a long cons tan t

B l i n e = f o r given s t e p s of A@. A cond i t iona l check is made t o note the

i n t e r s e - t i o n of t h e given r/a curve (denoted by a), and t h e va lues a t

t h e poin t of c ros s ing a r e determined by l i n e a r i n t e r p o l a t i o n . Since

r /a is multi-valued, a n a d d i t i o n a l check i s necessary t o note when t h e

inc reas ing va lues of r/a start t o decrease a t $I = nn/2, n = 1, 5 ... (denoted by C!), or when decreasing values start a t @ = nn/2, n = 3,

The

- ... . In t h i s manner, so lu t ions f o r many va lues of r/a can be obtained

f o r a gi7;en value of B.

Solu t ions f o r %/a and v/a3 versus ./a. These s o l u t i o n s were some-

what more d i f f i c u l t t o ob ta in than those descr ibed above s i n c e it w a s

nezessary t o increment @ as we l l as @. Both of t h e cond i t iona l checks

descr ibed above f o r c ros s ing a given va lue of r/a and a change from in-

c reas ing t o decreasing r/a values were requi red as w e l l as a check f o r

t h e change from inc reas ing t o decreasing values of h/a (denoted by A i n

F ig . 2 ) .

9

ORNL-DWG 71- 7610

Fig. 2. Schematic Representation of the Relationships Between h/a, p, r/a, and gr.

10

For a given va lue of f3, h/a i s a multi-valued f 'unction so that t h e

choice of i nc reas ing or decreas ing B t o approach h/a must be c a r e f u l l y

considered.

g iven va lue of r /a must a l s o be c a r e f u l l y considered.

most t roub le - f r ee s o l u t i o n over t h e e n t i r e h/a versus (b f i e l d , a

decremental approach from r i g h t t o l e f t (as shown by t h e arrows i n

F ig . 2) w a s chosen.

lQrthermore, t h e d i r e c t i o n of approach t o g/a a long t h e

To insu re t h e

To reduce t h e number of i t e r a t i o n s requi red t o l o c a t e x/a, an e s t i -

mate of t h e va lue of B i s ca l cu la t ed from equat ions f i t t e d t o a few

pre l iminary r e s u l t s . The i n i t i a l f3 value is then decremented a long

f i r s t a coarse g r i d , and f i n a l l y a long a n e x t r a f i n e g r i d t o o b t a i n t h e

f i n a l s o l u t i o n us ing double p rec i s ion .

Range, Running Time, and Accuracy

The ranges of t h e computer program as p r e s e n t l y w r i t t e n a r e

0 < $I < 3600, 0.1 4 r/a 4 2.0, and 0.02 4 f3 4 150; however, t h e s e can

be e a s i l y extended a t some s a c r i f i c e of e i t h e r t h e running t ime o r

accuracy.

of :/a f o r a given va lue of r/a i s approximately 10 seconds on t h e

II3M 360-91 Computer system.

programs i s cons iderably s h o r t e r pe r s o l u t i o n .

The average running t i m e f o r t h e program t o o b t a i n a va lue

The average running t ime f o r t h e o t h e r

An es t imate of t h e accuracy ( t o be given l a t e r ) w a s obtained by

comparing t h e values of h/a as a func t ion of @ f o r var ious va lues of fi,

A@, and f o r s i n g l e and double p rec i s ion .

The r e s u l t s of t h e computer s o l u t i o n a r e discussed i n t h e next

s e 1: t i on.

RESULTS

The r e s u l t s i n bo th t a b u l a r and g raph ica l form a r e presented i n

t h i s s e c t i o n and i n t h e Appendix.

The p r o f i l e of a bubble f o r f3 = 0.8 is shown i n F ig . 3. The pro-

f i l e is shown extended t o @ = 3600, which would be poss ib l e i f a s u i t a b l e

11

ORNL-DWG 74-7614 4.2

4.0

0.9

0.8

0.6 3

0.4

0.2

0 t.0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 i .0

x/b

Fig. 3 . Profile of a Bubble w i t h p = 0.8 for 0" 5 @ 5 360".

12

fo im of attachment w t ~ e provided, and the cen te r of bouyancy were t o

rennin on the z axis.

A t a I )u la t ion of B, X/II, z/b, fi/u, unci V/U' f o r variou:; values of $I

:in' pwsen ted i n 'hbles 13.1 through B.6 ( i n Appendix B). and V/aJ vcrsus

Figure IC c l e a r l y shows the a t ta inment of a mwclmum value of h/n (c/a) for a given radius of attachment.

p ressure technique" for t h e measurement of su r face t ens ion . )

t h e r e is a minimum value of h/a as wel l as a maximum value for V/a".

Plot:; o f h/a

for w r i o u s r/a a r e shown i n F igs . ) c and 5, r e spec t ive ly .

(This i s t h e b a s i s of t h e mximum-bul~ble-

I n add i t ion ,

The values of K/a and t h e corresponding values of v/a", B, 5, ;/I), - z/b for various values of r/a a r e given i n Table 1, and var ious c ross -

p l o t s of t hese v a r i a b l e s a r e given i n F igs . A . l through A.j+ (Appendix A ) .

These p l o t s show t h a t bo th K/a and 5 approach asymptotic values of r f i and 180°, r e spec t ive ly . Thus, t h e maximum pressure d i f f e rence

(x/a) t h a t a l a r g e tube can s u s t a i n w i l l be very nea r ly independent of

tube diameter.

*

A leas t - squares , polynomial f i t of t h e computer s o l u t i o n s for v a r i -

ous formulat ions of E/a and r/a were made.

t h e b e s t fit was

The forumlat ion t h a t gave

a a

h r h a

which i s of t h e same form as t h e pe r tu rba t ion s o l u t i o n s of Rayleigh-

Schr6dinger.

by the r e l a t i o n s h i p :

For t h i s reason, Eq. (19) was f i t t e d t o t h e da t a of Table 1

f ( y ) - io + i l y + i2Y Y3

F(Y) = = i3 + i,y + i5f + ... , (20)

where i 0'

s o l u t i o n and i,, iq, is, ... were determined by t h e leas t - squares pro-

cedure and a r e l i s t e d i n Table 2.

i,, and i, a r e the coe f f i c i en t s of t h e Rayleigh-Schradinger

Y

* These values can be obtained by t h e s o l u t i o n of Eq. (3) with

l/r, = 0 and l / b = 0.

ORNL-DWG 74-7612 6

5

4

$ 3

2

n v 0 40 80 120 4 60 200 240 280 320 360

+r

Fig . 4. Varia t ion of the Dimensionless Pressure Difference, h/a, as a Function of & for Various Values of B and the Dimensionless Radius of Attachment, r/a.

14

ORNL-DWG 71-7613 !02

5 c I I I I I I I I

2

f 0’

5

2

loo

5 m

4 0-’

5

[ [ 1 !,,LOCI OF r /u3 I

2

5

2

0 50 100 150 200 250 300 350 +r

Fig. 5. Variation of the Dimensionless Volume, V/a3, as a Function of Or for Various Values of the Dimensionless Radius of Attachment, r/a.

Table 1. bhximum Values of t h e Pressure (?;/a) and Corresponding Values of t h e S ize , Shape, and Volume of Bubbles as a

Funct ion of t h e Radius of Attachment (./a)

0.12 0.14 0.16 0.18

90.646 0.0290810

91.427 0.0521086 91.906 0.0662774

90 795 0 0397216 0.995156

0,988791

0.993412 0.99121r-3

0 999705 0.998120

1.00649 1 .OOY?97

o .003651 0.005806 0.008787 0.012624

92.411 0.082286 92.906 0.100192 93.420 0.120066 93.945 0.141985

0.986012 0.982929 0.979526 0.975814

1.00865 1.00988 1.01064 1.01077

5 J34652 4.693892 4.328985 4.022445

0.20 0 .?2 0.2)+ 0.26

0.01 7439 0.02?404

0.039223 0.030656

0.28 0.30 0 - 325 0.350

94.488 0.166037 95.474 0.192565 96.472 0.229022 97.476 0.269500

0.971786 0.966824 0.960416 0 * 95 3463

3 761790 3.537926 I: .299450 3 097815

0.049570 0.062105 0.080293 0.10191 1

1.01039

1.01785

1.01574 1.01752

98.987 0.315035 100.225 0.365084 101 .985 0.421688 io? .483 0.483756

0.94485 9 0.936222 0.925568 0.9149520

2.925744 2 777713 2.649595 2.5 38105

0.128876 0 ~ 5 8 4 7 5

0.2 39818 0 197373

0 * 375 0.400 0.425 0.450

1.02327 1.02319 1.02746 1.. 02640

0.475 0.50

0.60 0.55

104.821 0.551689 107.164 0.631617 111.986 0.822982 117.488 1.072133

0.904402 0.889730 0.857398 0.819487

2.440625 2 -355273 2.214232 2.104842

0.286173 0.346254 0.501889 0.708170

1.02173 1 .02464 1.02159 1.00938

123.990 1.407182

137.489 2.414075 130 993 1 * 853431

140.664 2.977900

0.986809 0.950582 0.902832 0 853059

0.65 0.70

0.80 0-75

0 77491-3

0 -682654 0.655617

0.727151 2.019914

1.902104 1.860445

1 * 95 3875 0.987762 1 347727 1.768822 2 ~ 3 6 1 6 3

152.491 4.841565 158.492 7.100181 162.989 10.070~60 165.994 13.866110

0.578448

0.490221 0.455742

0 530738 0.742939 0.648972 0.567348 0.497922

1.798649 1 75 3511 1.718720 1.690847

3.270626 4.356567 5.521721 6.736407

0.90 1 .oo 1.10 1.20

167.994 18.715220 169.991 25.16100 171.489 33.48783 172.494 44.16429

0.424972 0.394711

0.340486 0 * 366575

0.438376 0.385323 0.339160 0.299033

1.667904 1.648640 1.632205 1.618007

8.005399

10.839000 12.356187

9.386485 1.30

1.50 1.60

1.40

173.488 58.14618 173.998 75 .94175 174 9 987 175.489 130.1581

99 9 7358

0.315285 0.292110 0.268978 0.247919

0.263383

0.180031

0.232455 0.204340

1.605608

1 584975 1.576295

1.594682 13 971.941 15.640580 17 453834 19.314144

1.70 1.80 1.90 2 .oo

w

16

Table 2. Coef f i c i en t s for t he Polynomial Equations

F i t t e d t o t h e Computer So lu t ion

i 9

i 10

ill

1 . 00000 -0.66667

-0.66667

0.03230

-5 9 52833

61.19134

-351.38141

1099.76625

-1-930 43994

1913.36384

-1003.22519

216.93848

-0.00090

1.04439

-0.47175

1.43283

- 4.5 9801

5.38228

-2.73720

0.51837

W

W

The main disadvantage of Eq. (19) i s t h a t a n i t e r a t i v e procedure i s necessary t o c a l c u l a t e a/G. w a s a l s o f i t t e d t o t h e computer r e s u l t s :

A s impler formulat ion, bu t l e s s accu ra t e ,

The c o e f f i c i e n t s f o r t h e polynomial f i t of Eq. (21) a r e a l s o l i s t e d i n

Table 2.

The accuracy of t h e s e two polynomial f i t s i s discussed i n t h e next

s e c t ion.

DISCUSSION OF RESULTS

An es t imate of t h e accuracy of t h e computer r e s u l t s and compari-

sons with previous r e s u l t s a r e presented i n t h i s s e c t i o n .

E s t i m t i o n of t h e Accuracy. Resul t s were obtained f o r h/a versus c$

f o r @ = 0.1 and 100.0 wi th t h r e e values of A@ = lo, 1/2O, and 1/4" us ing

bo th s i n g l e and double p rec i s ion .

seventh decimal p l ace ) as &$ w a s decreased from 1' t o 1/4" when double

p r e c i s i o n was used f o r B = 0.1 and only 0.00008~ change f o r B = 100.0.

The s i n g l e p r e c i s i o n r e s u l t s a r e p l o t t e d i n F ig . 6, where t h e percent

d i f f e r e n c e is w i t h re fe rence t o t h e double p rec i s ion , A@ = 1/4", values .

A s can be seen, a n i n t e r v a l l e s s t han A@ = 1" decreased t h e accuracy of

t h e r e s u l t s because of rounding e r r a r s when s i n g l e p rec i s ion was used.

There w a s no change i n h/a ( t o t h e

Except f o r t h e maximum pressure r e s u l t s l i s t e d i n Table 1, a l l t h e

t abu la t ed r e s u l t s were computed us ing s i n g l e p r e c i s i o n and i n t e r v a l

= 1". The va lues l i s t e d i n Table 1 were ca l cu la t ed us ing double pre-

c i s i o n and are good t o a t least t h e sevnth decimal p lace . The o ther

t a b u l a t e d r e s u l t s a r e good t o a t l e a s t t he f i f t h decimal p lace .

Comparison wi th Previous Resul t s . A s a n t i c i p a t e d , t h e c a r e f u l ,

t ed ious hand c a l c u l a t i o n s of Bashforth and Adam (which requi red a number

of years t o complete) were i n good agreement ( t o t h e f i f t h decimal p l ace )

wi th t h e present computer s o l u t i o n . Trouble, however, develops when t h e

Bashfor th and Adam tables a r e s i n g l y and doubly i n t e r p o l a t e d t o apply

0.005

0

s z

V z W Q

-0.005

e -0.oIo LL

$ -0.045

0 I-

V 0: W (I.

-0.020

- 0.02 5

ORNL- QWG 74 - 7644

0 50 400 450 200 250 + Fig . 6. Comparison of the Single Precision

Double Precision, A$ = 1/4, Values as a Function

A+ 6 1.00 0.t 0.25 0.4

4.00 400

0.50 f00

0.25 400 300 350

Results for h/a with the of $7 a, and B.

t h e i r r e s u l t s t o p r a c t i c a l c a l c u l a t i o n s . Sudgen," "by c a r e f u l i n t e r -

po la t ion" of Bashfor th ' s tables, cons t ruc ted a table f o r c a l c u l a t i n g

s u r f a c e t e n s i o n by the maximum-bubble-pressure method.

t h e one most o f t e n r e f e r r e d t o i n cu r ren t l i t e r a t u r e on su r face t e n s i o n . )

The percent d i f f e rence between our s o l u t i o n and Sudgen's as a func t ion of

r/a is shown i n F ig . 7.

(This t a b l e i s

The maximum d i f f e rence i s -0.1%.

Although a n e r r o r of 0.1% can be neglected f o r some s t u d i e s a t

e leva ted temperatures (where o ther e r r o r s a r e more s i g n i f i c a n t ) , this

Magnitude of e r r o r can be s i g n i f i c a n t f o r m n y measurements made a t room

temperature , where t h e o r e t i c a l s t u d i e s of small changes i n t h e molecular

s t r u c t u r e of the i n t e r f a c e are be ing conducted. Futhermore, t h i s e r r o r

can be magnified by as much as 20 times when t h e two tube, d i f f e r e n t i a l

technique i s used t o measure sur face t ens ion .

To be p a r t i c u l a r l y noted i n F ig . 7 is t h a t t h e Rayleigh-Schradinger

s o l u t i o n is i n be t te r agreement w i t h t h e computer s o l u t i o n than Sudgen's

r e s u l t s a l l t h e way from 0 5 r/a 4 0.45. l a r g e p o r t i o n of t h e sur face t e n s i o n s t u d i e s that have been conducted i n

t h e p a s t . I n f a c t , t h e Rayleigh-Schradinger equat ion i s i n e r r o r by l e s s

t h a n 1.05% a l l t h e way t o r/a = 1.0.

s o l u t i o n can be used i n many cases where p r e c i s e su r face t e n s i o n values

a r e not needed.

This range of r/a covers a

Thus, t h i s much s impler a n a l y t i c

Also shown i n F ig . 7 a r e t h e devia t ions of Eqs. (19) and (21) from t h e computer s o l u t i o n s .

way t o r/a = 1.5. 0.2 S '/a 4 1.5; b u t Schr&TLnger's equat ion is recommended f o r r/a < 0.2.

The main disadvantages of Eqs. (1.9) and (21) i s t ha t double p r e c i s i o n

should be used i n t h e i r so lu t ion , e s p e c i a l l y a t t h e l a r g e r va lues of r/a.

Equation (19) agrees t o wi th in +O.O5% a l l t h e

The s impler Eq. (21) agrees t o wi th in +G.O7% from

Baumeister and H a m i l l p resented t h e i r r e s u l t s as p l o t s of d rop le t

volumes and he ights as func t ions of d rop le t r a d i i and con tac t ang le s .

I n t h i s form, t h e i r r e s u l t s could not be convenient ly compared w i t h t he

present r e s u l t s .

t h i r d s i g n i f i c a n t f i g u r e .

I n add i t ion , t h e i r r e s u l t s were given t o only t h e

20

ORNL- DWG 71 - 76 I S 0.5

0.4

0.3

0.2

0.1

0

-0.i

- 0.2

I I 1 I \ I I I I -

RAYLEIGH -SCHRODINGER

- EQUATION (is) - EQUATION ( 2t 1

0 0.2 0.4 0.6 0.8 4.0 i.2 I .4 4.6 4.8

do

1.4

4 .O

0.9

0.8

0.7

0.6

0.5

0.4

Fig. 7. Deviations of Previous Solutions and Present Polynomial Equations from the Present Computer Solution of the Maximum-Bubble Pres sure.

21

CONCLUSIONS

A computer program was w r i t t e n t o so lve t h e second-order, nonl inear ,

d i f f e r e n t i a l equat ion desc r ib ing t h e a x i a l l y symmetric i n t e r f a c e between

two immiscible f l u i d s . The present s o l u t i o n i s l i m i t e d t o sess i le -shaped

drops and bubbles; t h e s o l u t i o n f o r pendant-shaped drops and bubbles

r e q u i r e s a d i f f e r e n t approach and w i l l be given i n a l a t e r r e p o r t . All of t h e present r e s u l t s a r e accu ra t e t o a t l e a s t t h e f i f t h decimal p lace

wi th some r e s u l t s accu ra t e t o the seventh decimal p lace .

The r e s u l t s a r e presented i n a form t h a t i s u s e f u l i n t h e a n a l y s i s

of b o i l i n g and condensing h e a t t r a n s f e r , superheat , c r i t i c a l cons tan ts ,

and i n t h e measurements of contac t angles and su r face t ens ion . I n a d d i -

t i o n , t h e s e r e s u l t s may be of use i n t h e design of equi -s t ressed s h e l l s

f o r containment v e s s e l s (i .e. , above-ground water tanks , under-water

s to rage of l i q u i d s and gases, l i g h t e r - t h a n - a i r ba l lons , and submerged

marine l a b o r a t o r i e s ) . The computer program i s l i s t e d so t h a t a wider

range and number of v a r i a b l e s can be obtained as des i r ed .

1. F. Kre i th , P r i n c i p l e s of Heat Transfer , pp. 308-437, I n t e r n a t i o n a l Textbook Company, Scranton, Pennsylvania, 1st ed., 1959.

2 . J. A. Edwards and H. W . Hoffman, Superheat Cor re l a t ion f o r Boi l ing A l k a l i Metals, Proceedings of t h e Fourth I n t e r n a t i o n a l Heat Transfer Conference, V e r s a i l l e s , September 1970 (under t h e book t i t l e "Heat Transfer 1970"), E l sev ie r Publ i sh ing Cc-npany, Amsterdam, Netherlands, 1-970

3 . A. V . Grosse, The Rela t ionship Between t h e Surface Tension and Energies of Liquid Metals and Thei r C r i t i c a l Temperatures, J. Inorg. Nucl. Chem., 24:147-156 (1962).

4. D. W. G. White, Theory and Experiment i n Methods for t h e P r e c i s i o n Measurement of Surface Tension, Trans. ASME, 55: 757 (1962).

5. Lord Rayleigh, On t h e Theory of t h e Cap i l l a ry Tube, Proc. Roy. SOC., 92 (Se r i e s A): 184-195 (1915).

6. Erwin Schrgdinger, Notizhrber den Kapi l la rdruck i n Gasblasen, Ann. Physik., 46: 413-418 (1915)

22

7.

8.

9 .

10.

A

a2

b

g

D -::

h

h -

i

K

m

P

r

r/a

r l ,

R

F. Bashforth and J. C . Adam, An Attempt t o Test t h e Theories of Cap i l l a ry Act ion by Comparing t h e Theore t i ca l and Measured Forms of Drops of F lu ids , Univers i ty P res s , Cambridge, Massachusetts, 1883.

K. J. Baumeister and T. D. H a m i l l , Liquid Drops: Numerical and Asymptotic Solu t ions of t h e i r Shapes, NASA-TN-D-4779, National Advisory Committee f o r Aeronaut ics , September 1968.

F. B. Hildebrand, In t roduc t ion t o Numerical Analysis , pp. 236-239, McGraw-Hill, New York, 1956.

S. Sugden, The Determination of Surface Tension from Maximum Pressure i n Bubbles, J. Chem. SOC., 1: 858-866 (1922).

NOMENCLATURE

d i s t ance from t h e o r i g i n t o t h e plane of attachment, c m

s p e c i f i c cohesion E 2yg /pg, a2

r ad ius of curva ture a t t h e ve r t ex of t h e bubble, cm

l o c a l a c c e l e r a t i o n due t o g rav i ty , cm/sec"

dimensional cons tan t , dyne. sec2/g.cm

C

pressure d i f f e r e n t i a l ac ross i n t e r f a c e , cm of f l u i d wi th d e n s i t y p

m a x i m value of h f o r given va lue of r/a, cm of f l u i d with dens i ty p

c o e f f i c i e n t s of polynomial equat ions

c o e f f i c i e n t s of numerical equat ions

c o e f f i c i e n t s of numerical equat ions

pressure d i f f e r e n t i a l ac ross i n t e r f a c e , dyne/cm2

rad ius of c i r c l e of a t tachment , cm

dimensionless r ad ius of attachment

r2 p r i n c i p a l r a d i i of curva ture of t h e i n t e r f a c e , cm

dimensionless value of r a t rz/b

W

V volume enclosed by t h e i n t e r f a c e , cn3

23

V

X

- X

X

‘r

Y

Z

- Z

Z

r Z

volume enclosed by t h e i n t e r f a c e f o r a given value of r/a a t h/a = h/a, cm3

horizontal coordinate , em

value of x f o r a given value of r/a a t h/a = h/a, cm

dimensionless x, x/b

value of X a t x = r

independent v a r i a b l e i n polynomial equat ion

V e r t i c a l coordinate , cm

va lue of z f o r a given va lue of r/a a t h/a = h/a, cm

dimensionless z, z/b

value of Z a t x = r

Greek L e t t e r s

B

B

Y i n t e r f a c i a l t ens ion , dyne/cm

dimens ion le s s parameter 5 gpb3/gcy = 2 (b/a)”

va lue of p f o r a given r/a a t h/a = h/a -

@

@r

‘r

angle between a x i s of r evo lu t ion and t h e normal t o t h e sur face

value of @ a t the r ad ius of attachment

va lue of (5r for a given r/a a t h/a = h/a -

P p o s i t i v e d e n s i t y d i f f e r e n c e between the two f l u i d s , g/cm3

APPENDICES

A. PARAMETRIC CROSSPLOTS

Various c rossp lo ts of t h e parameters shown i n Figs. 4 and 5 a r e

given i n Figs. A.l, A.2, and A.3. The maximum bubble radius as a funct ion of t he angle $I for various r a d i i of attachment i s shown

p lo t t ed i n Fig. A.4. r be t h e radius of attachment.

t h e maximum bubble radius where the maximum bubble pressure i s

reached.

r For 9 < go*, t h e maximum bubble radius would

Also shown i n Fig. A.4 is t h e l o c i of

27

ORNL- DWG 74- 7646 9

8

7

6

5 El0

4

3

2

i 0 0.2 0.4 0.6 0.8 i.0 4.2 1.4 i.6 1.8 2.0

‘10

Fig. A.l. Interface Separating Two Fluids as a Function of the Dimensionless Radius of Attachment.

Dimensionless Maximum-Pressure-Difference Across the

V

ORNL-DWG 71-7647 80

70

60

50

a 40

30

20

40

0 90 400 440 420 430 440 450 460 470 480

5 r Fig . A.2. The Value of t he Parameter @ at h/a = x/a as a Function of @ .

r

ORNL- DWG 74- 7618

Fig. A . 3 . The Dimensionless Radius of Attachment as a Function of % at h/a = h/a.

ORNL-DWG 7!-7649 2.2

2.0

t.8

4.6

4.4

X

5 a 4.2

3 Y

1.0

0.8

0.6

0.4

0.2

0 90 t20 450 480 240 240 270 300 330 360

+r

Fig. A . 4 . bkximum Radius of a Bubble (x/a at @ = 90") as a Function of GT for Various Radii of Attachment (r/a). For $r C go", (x/a)max = r/a.

B. TABULATED RESULTS

The computer r e s u l t s f o r t h e s i z e , shape, pressure , and volume

of a t t ached bubbles and drops for given r a d i i of attachment are l i s t e d

i n Table B.l through B.6. The Results a r e arranged i n ascending va lues

of @r*

33

34

Table B.l. Size, Shape, Pressure, and Volume of Bubbles and Drops with a Given Radius of Attachment

r/a = 0.2

30 .oo 18.00 8.00 5 -50 4.00

3.50 2.90 2.30 2.00 1.80

1.50 1.10 0.95 0.80 0.66

0.58 0.50

0.38

0.34 0.30 0.25 0.23 0.21

0.15 0.17 0.15 0.14 0.13,

0 .l2 0.11 0.10

0.46 0.42

0.09

2.99 3.86 5.80 7.00 8.21

8.78 9.66 10.86 11.66 12.30

1-3.49

17.05 18.63 20.55

15.81

22.04 23.84 24.92 26.17 27.62

29-35 31.46 34 87 36.59 38.60

40 59 43 91 47.60

52-51 49.86

55.69 59.65 64.87 72 * 70

0.0828 84.85

0.05168 0.06667 0.10000 0.12061 0.14142

0.15119 0.16609 0.18650 0.20000 0.21082

0.26968 0.29019 0.31623 0.34816

0.23094

0 37139 0.40000 0.41703

0.45883 0.43644

0.48507 0.51640 0.56569 0.58977 0.61721

0.64889 0.68599 0.73030 0.75593 0.78446

0.81650 0.85280 0.89443 0.94281 0 98295

0.00135

0.00505 0.00734

0.00225

0.01013

0.01158

0.01765

0.02262

0.01400

0.02317

0.02717

0.04327 0 - 03727 0.05161 0.06295

0.08399 0.09166 0.10089 0.11220

0.12636

0.17662 0.19386

0.07194

0.14463

0.21486

0.24102 0.27474 0 31993 0.34889 0.38410

0.42796

0.56306 0.68614

0.48479

0. a8466

0.26341

0.61519

0.34009 0.51009

0.72143

0 -77125 0.84731

1.02032

1.17823 1.37604

1.61378 1.77694

1.89569

2.12910

2.34306

2.47745 2.63801 2.89087

0.95144

1.07556

1.48078

2.04200

2.22841

3.01458 3 *15569

3 31872 3 51007 3 * 7393-0 3 87195 4.02025

4.18731 4 37771

4.85960 5 09473

4.59804

0.00030

0.00063

o.00090

0.00096 0.00106 0.00119 0.00128

0.00042

0.00077

0.00135

0.00149 0.00175 0.00189 0.00207 0.00229

0.00268 0.00281 0.00296

0.00247

0.00313

0.00334

0.00405 0.00427 0. GO45 5

0.00489 0.00531

0.00361

0.00585 o .00622 0.00664

0.00719 0.00792

0.01079 0.01444

0.00898

35

Table B .1 (Continued)

0.0828 0.09 0.10 0.11 0.12

0.13

0.15 0.17 0.19

0.23 0.25

0.38

0.46 0.58 0.66 0.70

0.80 0.90 1 .oo 1.10

1.10 1 .oo

0.80

0.14

0.21

0.34

0.42

0.90

95 -17 107.58 115.80

125.82

129.46 132 59 135 34 140.07 144.07

121.43

147.60 150.78

169.34

173 53 177.57 189.15 196.74

1-53.70 164.94

200.55

210.35

233.12 220.90

250.16

306 55 289.74

318.54

0 9 98295

o .85280 0.81650

0.94281 0.89443

0.78446 0.75593 0 73030 0 68599 0.64889

0.61721 0 58977 0 56569 0.48507 0.45883

0.43644 0.41703 0 37139 0.34816 0.33806

0.31623 0.29814

0.26968

0.26968 0.28284 0.29814

0.28284

1.05331. 1.24471 1.35781 1.42604 1.47280

1.50654 1.53162 1.55053 1 57565 1 58951.

1.59609

1.59596 1.56461

1.59777

1.54451

1.52274 i.50000 1.42956 1.38256 1.35938

1.30143 1.24333

1.11197 1.18282

1.01358

1.01038 1.00345

5 -12905 4 97809 4 9 77575 4.59845

4.30642

4.07611 3 88935 3 73435

3.60326 3.49067 3.39269 3.07046 2 96739

2.87998 2.80452 2.62679 2 53500 2.49453

2.40423 2.32476 2.25059 2 ~7306

2.10009 2.12376 2.16850

4.44324

4.18487

0.01878 0.02660

0.04890

0.05606 0.06328 0.07058 0.08542 0.10060

0.11610 o .13189

0.25668

0.32574

0 49949 0.53405

0.61955 0 070350 0 * 78543 0.86409

0.85531

0.68854

0.03445 0.04173

0.14794 0.22259

0.29109

0.43005

0.77164

328.88 0.31623 1.02769 2.23111 0.60513

Table €3.2. Size, Shape, Fressure, and Volume of Bubbles and Drops with Given Radius of Attachment

r/a = 0.3

30.0 18.0 12 .o 8.0 5.5

4.0 3.5 2.9 2.3 2.0

1.8 1.7 1.5 1.25 1.2

1.1 1 .o 0.95 0.9 0.8

0.7 0.66 0.58 0.5 0.46

0.42 0.38 0.34 0.3 0.25

0.23 0.21 0.195

4.55 5 *87 7.20 8.83 10.66

12 53 13.41 14 77 16.63 17.87

18.88 19.45 20.76 22.85 23 35

24.46 25.74 26.46 27.24 29.05

31.28 32 * 32 34 79 37 92 39 86

42.13 44.87 48.25 52.62 60.61

65 36 72.20 81 53

0.07746

0.15000 0.18091

0.21213 0.22678 0.24914 0 -27975 o.30000

0.31623

0.34641 0 37947 0 38730

0.10000 0.12247

0.32540

0.40452 0.42426

0.44721 0 43529

0.47434

0 50709 0.52223 0 55709 o.60000 0.62554

0.65465 0.68825 0.72761 0.77460 0.8485 3

0.88465

0 96077 0.92582

0.00307 0.00509

0.01145 0.00764

0.01672

0.02640

0.04663

0 9 05197 0.05512 0.06271 0 -07579 0.07908

0.08662 0 09574 0.10110 0.10704 0.12135

0.14017 0.14943

0.20334

0.02303

0.03194 0.04045

0.17222

0.22365

0 27994 0.32082 0 37669 0.48659

0 55577 0.65884 0.80327

0.24857

0.27011 0 34859

0.63076

0 73967 0 9 79085 0.86891 0 97588 1.04663

1.10340 1 13547 1.20901 1.32483 1 9 35225

1.41264 1.48191

0.42700 0.52291

1.52063 1.56252 1 65789

1 77323 1.82662

2.10167

2.29609 2.41618 2 55764 2.72788 3.00046

3 13731 3 29956 3 45338

1.9497b

2.19240

0.00165 0.00215 0.00264

0 900393

0.00464 0.00496

0.00618 0.00667

0.00325

0.00547

0.00745 0.00727 0 00779 0.00861 0.00881

0.00925 0 00977 0.01005 0.01038 0.01113

0.01206 0.01250 0.01358

0.01588

0.01695 0.01829 0.02000 0.02236 0.02715

0.03036 0.03559 0.04423

0.01499

37

Table B.2 (Continued)

B @ x/b z/b h/a v/a3

0.1925 0.21 0.23 0.25 0.3

0.34

0.42 0.46

0.38

0.50

0.58 0.66 0.7 0.8 0.9

1.0 1.1 1.2 1*5 1.7

1 *7 1.2 1.1 1 .o 0.9

95.40

121.34 131.17

137.09 142.08 146.46 150.41 154.05

108.40 115.90

160.65 166.63 169.45 176.18 182.56

188.74 194 79

219.70 234.58

304.88

200.82

337.48 343 30 349 15 355 -15

0.96694 0.92582 0.88465

0.77460

0.7276 0.68825 0.65465 0.62554 o.60000

0 55709 0.52223 0 50709 0.47434 0.44721

0.8485 3

0.42426 0.40452 0 38730 0.34641 0.32540

0.32540

0.40452 0.42426 0.44721

0 9 38730

1.01465 1.19288 1 27945 1.33222 1.40212

1.42704 1.43800 1.44035 1 - 43727 1.43052

1.41014 1.38465 1.37088 1- 33487 1.29805

1.26124 1.22489 1.18910 1.08403 1.01218

0 e83893 o ,87818 0.89730 0.92136 0 95155

3 9 53793 3.47260 3 38272 3 9 29944 3 12503

3.01374 2 92095 2.84223 2 ’ 77443 2.71526

2.61634 2.53620 2 50133 2.42538 2.36147

2.30604 2.25680 2.21206 2 09350 2.01784

1.85810 1 971-23 2.01386 2.06571 2.12903

0.06203 0.08758 0.10864 0.12796 0.17414

0.21045 0.24666 0.28285 0 31905 0 35524

0.42743 0.49919 0.53471 o .62303 0.70984

0 79517 0.87863 0.96038 1.19399 1 33856

1.29849

0.84032 0 91837

0.76142 0.68168

Table B.3. Size, Shape, Pressure, and Volume of Bubbles and Drops with Given Radius of Attachment

r/a = 0.4

30.0 18.0 12 .o 8.0

5 *5 4.0 3.5 2.9

2.3 2 .o 1.8 1 . 5

1.25 1.1 1.0 0.8

0.66 0.58 0.5 0.46

0.42 0.38

0.36 0.38 0.42 0.46

0.5 0.58 0.66 0.7

0.8 0.9 1 .o 1.1

6.17 7.98 9-79

12.02

14.54 17-13 18 35 20.24

22.86 24.63 26.06 28.78

31.84 34.23

41.32 36.17

46.69 50 - 97 56.89 60.93

66.32 74.69

97.25 105 99 115.44 121.96

127.16 135.45 142.15 145.12

151.82 157 75 163.17 168.21

0.10328 0.13333 0.16330 0.20000

0.24121 0.28284 0 * 30237 0.33218

0 37300 0.40000 0.42164 0.46188

0.50596 0.53936 0 56569 0.63246

0.69631 0.74278 0.80000 0.83406

0,87287 0.91766

0.94281 0.91766 0.87287

0.80000

0.69631 0.67612

0.63246 0.59628 0 56569 0.53936

0.83406

0.74278

0.00548

0.01372 0.00912

0.02063

0.03019 0.04173 0.04787 0.05808

0 07385 0.08546 0.09548 0 a 5 8 9

0.14108 0.16228 0.18038 0.23257

0.29255 0.34404 0.41997 0.47434

0.66956

0 98565 1 * 09093 1.18463 1.23412

1.26371

1.29953 1.29854

0.54940

1.12924

1.28804 1.27074 1.24977 1.22692

0.27941

0.44187 0.36068

0.54126

0.65309 0.76612 0.81926

1.01170 1.08546

0.90040

1.14468 1.25507

1.37644 1.46875 1.54176 1.72823

1.90883

2.20998 2.31263

2 943395 2.58601

2.04222

2 77520 2.76968 2 72505 2.67701

2.63185 2 55291 2.48730 2.45854

2 39577 2 34315 2 29794 2.25831

0.00533 0.00692 0.00850

0.01270

0.01613 0.01786

0.02030 0.02196

0.01047

0.01504

0.02332 0.02594

0.02897 0 -03139 0.03 341 0 03897

0.04512 0.05041 0.05828 0.06413

0.07262 0.08785

0.14837 0.18416 0.23502 0 -27950

0.32143 0.40169 0 47925

0.61073

0 51723

0.70194 0.79124 0.87839

39

Table B.3 (Continued)

1.2

1.5 1.7

1.25

2.0 2.5

172.98

186.20 175 -29

194.51

206.81 229.01

310.00

342.67 350 54

331.06

0.51640 0.50596 0.46188 0.43386

0.40000 0 35777

0 - 35777 0.40000 0.43386 0.46188

1.20312 1.19105 1 * 13075 1.08367

1 01557 0.90542

0.71 442 0.73401 0.76316 0 79225

2.22293 2.20652 2 9 13396 2 08375

2 01557 1.90671

1.69317

1.78825 1 ,73401

1.84081

0 96372 1.00587 1.20846 1.36230

1.57998 1 * 90695

1.81377 1.47971 1.27326 1 A3175

40

Table 13.4. Size, Shape, Pressure, and Volume of Bubbles and Drops with Given Radius of Attachment

r/a = 0.6

60.0 30.0 18.0 12 .o 8.0

5.5 4.0 3.5 2.9 2-3

2.0 1.8 1.7 1.5 1.25

1.2 1.1 1 .o

1 .o 1.1 1.2 1.25 1.5

1-7 1.8 2.0 2.2 3.0

3.5 4.0 6.0

6.0 4.0 3.0

6.98 9-75 12.64 15.54 19.17

23 * 35 279 72 29.83 33.16 37.95

41.31 44.15 45.82 49.88 57.15

59 11 63.92 70.86

110.89 119.62 126.22 129.07 140.75

148.28 151.67

166.26 183.08

157 90

1-93-87 204.25 258.64

281.23

., 251.28 J 32.13

0.11094 0 ~5492 0.20000 0.24495 o.30000

0.36181 0.42426 0.45356 0.49827 0.55950

o.60000 0.63246 0 65079 0.69282 0.75895

0.77460

0.84853

0.84853

0.77460 0 9 75895 0.69282

0.63246 o.60000 0.55950 3.489%

0 -45356 0.42426 0.34641

0.34641 0.42426 0.48990

0.80904

0.80904

0.65079

0.00647 0.01266 0.02120 0.03198 0.04838

0 907133 0 09970 0.11501 0.14101 0.18242

0.21408

0.25954 0.30298 0.38608

0 40935 0 46757 0.55331

0 97076

1.04278

1.05581

1 03679

0 98579 0.90845

0.85574 0.80597 0.60326

0.55639 0 55032 0 59863

0.24231

1.01944

1.04934

1.04488

1.01784

0.21801

0 39692

0 59676

0.30724

0.48658

0.72131 0.84811 0.90807 1.00025 1.12813

1.28397 1 32394

1.21407

1.41709 1.57014

1.60808 1.69516 1.80546

2.10065

2 09873

2.06906

2.04798 2 9 37679 2.01784 1.98964

1.88796 1.84692 1.62222

1 48537 1.54966

2.10444

2.09449

1.92912

1.54104

0.01748 0.02813 0.03656

0.05604

o .06879 0.08255 0.08935 0.10029 0.11668

0.12865

0.16135 0.19222

0.04515

0.13910 0.14544

0.20112 0.22420 0.26115

0.61473

0.85283 0.90534

1.32506

1.57280 1.80255 2.28289

2 9 58703 2.86294 3.68271

3 9 59171 2.56096 2 003835

0.74147

1.14754

1.40998

41

Table B.5. Size, Shape, Pressure, and Volume of Bubbles and Drops with Given Radius of Attachment

60.0 30.0 18.0 16.0

12 .o 8.0 6.5 5-5

4.5 4.0 3 -7

3.7 4.0 4.5 5.5

6.5 8.0 10.0 16.0 20.0

20.0 16.0 10.0

13.46

25.24 26.91

31.60 40.18 45 96 51-73

61.02 69.09 77 81

103.15 113 9 93 125 33 140.90

152.52

19.24

166.51 181.82 220.40 253.01

286.54 316.08 348.47

0.18257 0.25820 0.33333 0 -35355

0.40825 o.50000 0 55470 0.60302

0.66667 0.70711 0 73522

0 73522

0.66667 0.60302

o.50000 0.44721 0.35355 0.31623

0.31623 0 35355 0.44721

0.70711

0 55470

0.01914 0.03877 0.06585 0.07454

0.10137 0.15888 0.20262

0.32813 0 9 39734 0.46919

0.63176 o .67052 0.69002 0.68546

0.66411 0.62550

0 36935

0.24929

0 57493 0.44901

0.32147 0.31169 0.35006

0.28739

0 53089 0.56438

0 65655 0.81776 0 91-998 1.01642

1.15886 1.26903 1 37338

1 59450 1.65536 1.70170 1 73973

1.75195 1.75100 1 73279 1.62354 1.48421

1.33281

0.40835

1.23514 1-22997

0.17182 0.24755 0.32807 0 35097

0.41635

0.63182 0.72663

0 -89237 1.05200 1.24390

1 95007 2.32888 2.78302 3.48414

0.54200

4.05442 4.76824 5 054361 7 13667 7 -66733

7 19870 6 905930 4.49208

42

Table 8.6' Size, Shape, Pressure, and Volume of Bubbles and Drops wi th Given Radius of Attachment

r/a = 1.5

100 .o 80 .o 60.0

30 .o 24.0 20.0 18 .o 16.0

18.0 20 .o 24.0 30.0

50 .o

80 .o

80 .o

50.0 32 .o

40 .O

64.0

64.0

21 .oo 0.21213 23.69 0 - 23717 27 71 0.27386 34.97 0 33541

41.80 0 38730

56-13 0.47434 61.81 o.50000 70.85 0 53033

48.69 0.43301

124.73 o.50000 133.98 0.47434

163.84 0.38730 148.02 0.43301

200.61 o.30000 222.48 0.26517 255.04 0 023717

284.50 0 -23717 313.14 0.26517 330.62 o.30000 354.65 0 37500

o ,03058 0.03859 0.05210 0.08054

0.11119

0.18271 0.21185 0 9 25677

0 39672 0.39444 0 38033 0 9 35409

0.23868

0.16827 0.16132 0.16778 0.19654

0.14478

0.27914

0.19292

0 35762

0 46793 0.58381

0,68884

0.89402 0.96887 1.07982

1- 52349 1 9 56354 1.60617 1 *62959

1 59568 1 52697 1.37824

1.22238 1.08935 1.03892 1.03617

0.40220

0.79022

0.83884 0.94928 1.11632 1.42593

1.72815 2.04596 2.40649 2 69503 3.18127

6.89629 7.66084 8.85724 10.20717

12 93770 13 97587 14.29483

13.20281 11.13856 9.65580 7 76366

C . COMPUTER PROGRllM

The F o r t r a n l i s t i n g of t h e computer program for so lv ing the i n t e r -

f a c i a l equat ion and computing t h e var ious parameters presented i n t h i s

r e p o r t i s given i n Table C.l. e a s i l y extended a t some s a c r i f i c e of e i t h e r t h e running t ime o r t h e

accuracy.

a func t ion of drop

modifying one of t h e MAIN subrout ines o r by adding another r o u t i n e .

The ranges of t$? B, and r/a can be

Other parameters, such as maximum drop he ight and r ad ius as

volume and con tac t angle , can be obtained e i t h e r by

43

44

Table C .1. Computer Program

C S O L U T I O N OF I N T E R F A C I A L EO. FOR T H E P R E S S . A N D VOL. C H I T H I N A T T A C H E D BUBBLE, CASE 1 - A T T A C H E D A B f l V E ( P O S I T I V E B E T A ) C

I F l P L I C I T REAL:z8 ( D ) T R E A L * ~ ( A - C T E-HTO-Z) D I M E N S I O N A ( ~ ) T B ( ~ ) T C ( ~ ) T D ( ~ ) , E ( ~ ) T R ( ~ ) T G ~ ( ~ O ) T P ( ~ ) T!I M E N S I O N D A ( 4 ) ,DB( 4 ) y D C ( 4 ) 9 DZ ( 4 ) 9 D E ( 8 T D P ( 4 ) T D R ( 4 )

9 FORPTAT ( 1 1 7 1 2 ) 11 R E A D ~ T I T N ~

I F (I -1 I 4 0 1 9 1 0 1 5 O 1 c, C S O L U T I O N A T MAX.PRESS FOR G I V E N V A L U E S OF R / A ( = G ) C

300 F O R M A T ( F 1 5 . 7 ) 10 C O N T I N U E

DO 2 0 1 N 4 = l r N 3 R F A D 3 0 0 , G

C E S T I M A T E OF I N I T I A L B E T A I F (G.GE.0.099.AND.G. L T . 0 . 4 ) G O T O l O O l I F ( G .GE -0 -39 . AND .G . L T .O .8 ) G O T 0 1 0 0 2 IF(G.GE.0.79 .AND.G . L T , 3 . 0 1 ) G O T 0 1 0 0 3

G O T 0 3 0 1

(I I! T 0 3 0 1

G O T 0 3 0 1 C CCIURSE S E T A G R I P

3 0 1 t3 E T A = B E T A+ 0 . 1 ;::R E T A 1\120=0 P (2)=0

C A L L I F ( P ( 1 1 -P ( 2 1 1 59 t 59 ~6 0

10 0 1 B E T A = 1 1 17 5 - 0 9 2 43 4:::G + 3 09 2 5 2 l * G *::: 2

1002 PETA=2.87267-12o78143:: ’G+16.40021*G*‘~’2o

10 0 3 B E T A = 1 6 3 0443-49 4 3 18 8 :::G +40 5 46 28 aG :::‘; 2

58 BETA=BETA-.Ol;: :BETA !+A I NS ( A 9 B T C ,Z Y E t H J t P T B E T A t G T R 1

6 0 P ( 2 ) = P ( 1 ) N 2 0 = N 2 0 + 1 GOT1158

59 57 P R I N T ~ O T B E T A T H

I F ( N 2 0 - 1 ) 301 t 30 1 t 57

P R I N T 100

B E T A = B E T A + . 02:::B E T A P ( 2 ) = 0

B E TA= B E T A-• 0 0 1’:: B E T A C A L L 143 I N S ( A I F ( P ( 1 1 -P ( 2 1

PR 1 N T 2001 E ( 8 T E ( 6 T E ( 7 TR ( 1 T P ( 1) T V

C F I N E B E T A G R I D 6 5

B t C T Z 7 E 7 H 9 J 7 P 7 B E TA TG TR 1 6 1 T 6 1 T 6 2

6 2 P ( i l ) = P ( l ) G O T 0 6 5

45

C

Table C .1 (Continued)

C C C

C C C

6 1 P R I N T 50 ,BETAYH P R I N T 100 P R I NT B E T A = R E T A+ . 0 0 2:;: B ET A DP ( 2 1 = o D B E T A = B E T A DG =G

200 ,E ( 8 1 ,E ( 6 ,E (7 1 ,R ( 1) ,P ( 1) V

EXTRA F I N E B E T A G R I D W I T H DOUBLE P R E C I S I O N

66 DBETA=DBETA- .OOOl ' : 'DBETA CALL M A 1 ND (DA,DB,DC, DZ DE, DH J DP DRETA, DG, D R ) I F ( DP ( 1 - DP ( 2 1 6 3 7 6 3 6 4

6 4 D P ( ? ) = D P ( l ) G O T 0 6 6

6 3 D V = 3 1 4 1 59 2 6 5 :::DG

D E ( 8 ) = D E ( 5 ) : : 5 7 . 2 9 5 7 7 9 5 7

+ ( DP ( 1 1 ':: DG - D SI N ( DE ( 5 ) ) ) D E ( 4 1 =DE ( 1 ) ::'57 029 5779 57

P R I N T 50,DBETA,DH 5 0 F t I R F \ A T ( l t i 2 r 6 H f i E T A =,E1507 9 6 X 73" = , F 1 5 . 7 / / / ) 100 F ( l R M A T ( l H O , 5 X , 3 H P I - I I , 8X ,3HX/B , 1 3 X , 3 H Z / B r 1 3 X , 3 H R / A , 1 3 X , 3 H H / A ,

113X , 4 l i V / A 3 FOR l4A T ( 1 H 2 00 OP I F 9 3 1P 5 E 16 e7 1 PR I N T 1 0 0

2 0 1 C O N T I NU€ P R I N T Z O O y D E ( 8 1 ,[)E((> ) , D E ( 7 ),DG ,DP (1) ,DV

GOTO 11

S U L I J T I O P ! FOR G I V E N V A L U E S C F BETA APJD R / A ( = G l )

400 F ( l R M A T ( 8 3 1 0 . 7 1 4 G 1 C O N T I PJUE

DO 203. N 4 = l , N 3 R E A D LOOrBETA, ( G l ( 1 ) 9 1 = 1 7 8 1 C A L L M A I N I S ( A ~ B , C ~ Z rE,H,J ,P ,BETA,Gl ,R 1

2 0 2 C O N T I NIJE G O T O 11

S O L U T I O N FOR G I V E N V A L U E S OF BETA A T I N T E R V A L S ( N 1 ) OF P H I UP T O N 2

5 0 1 C O N T I NUE DO 203 N 4 = l r N 3 REA C 500 B E T A ,N 1, N 2

500 F i I R M A T ( F 1 0 . 7 r 2 1 4 - 1

203 C O N T I N U E GOTO 11 END

C A L L MA I 1\12 S ( A B C ,Z E ,H J P B ETA G I? N1, N 2 1

C C C

46

Table C .1 (Continued) -

SURRfIUTINE MAINS(A,B,C,Z,E,H,J*P,BETA,G,R) c

IMPLICIT REAL::::, (D1 ,REALx<4(A-C, E-H,O-I) DIMEf\JSIfl~AA~~),R(4),C(4),Z(4),E(8),P(4),R(4) H=O. 174532?E-01 E(b)=O. J=3 E(l)=(-). E(2)=0. E(3)=0. Nl=l R(2)=0. D0150M=l,360 CALL RlJNGKS(A,Ec,C,Z,E,H,J,BETA,G,P) R(l)=E(2):::S.7RT(BETA/2.0) GOTO(4v51,Nl

4 IF(G-R(1))5,3,3 3 IF(R(l)-R(2))8,8,10 5 IF(G-R(1))1,6,7 1 N1=2

10 RtZ)=R(l) E(5)=E(l) E(6)=F(2) E(71=E(3)

150 CnNTI b!LlE 7 F=(G-R(2))/(R(l)-R(2))

F(M)=F,::(E(l)-E(5))+E(5) E(h)=F,::(~~(2)-E(6))+E16) ~(7)=Fi:(f(3)-E(7))+E(7)

h P(1)=S(-~RT(%,/EETA)-tE(7),::SQRT(BETA/2.) GOT09

8 P(ll=l.O P(2)=2*0

9 2 Fj T (JR i\i Fr i< I‘,

c c C

SUBRT)UTINE MAIND(DA,DB,DC,DZ,DE,D+l,J,DP,DBETA,DG,DR) c

IMPLICIT REAL:::B(D),REAL+4tA-C,E-H,O-Z) DIME~iSIC3~DA(4),DrJ(4.),DC(4),DZ (4),DE(8)tDP(4),DR(4) Dl-i = 0 . ;? 7 26646 D-02 DE(6)=0. J=3 DE(l)=O. DE(2)=0. DE(3)=0. ?dl=l

.

-

47

Table C .1 (Continued)

D015OM=l,720 CALL RUNGl<D(DA,DB,DC,DZ,DE,DH,J,DBETA,DG,DP, TJR(l)=DE(.?)~~l)S~?RT(DRETA/Z.O) GOT0(4,5),Nl

4 IF(DG-DR(1))5,150,150 5 IF(DG-DR(lf)1,6,7 1 DR(2)=DRIlI

DE(5)=DE(l) DE(6)=DE(2) DE(7I=DE(31 N1=2

150 CONTINUE 7 DF=(DG-DR(Z))/(DR(l)-DR(2))

DE(5I=DF~:(DE(l)-DE(5Il+DE(5) I)E(6)=DF:::(DE(Zl-[)E(b))+DE(6) DE(7)=DF:~(DE(3)-DE(7))+DE(7)

6 DP(11=DSQRT(2./DBETAI+DE(7)~~:DSORT(DBETA/2.) RETURN END

L

SUBROLJTINE NAINISIA,B,C,Z ,E,H,J,P,BETA,Gl,R) c

IMPLICIT REAL:% (O),REAL:::rt(A-C,E-ti,O-Z 1 DII,iEi\lSIOi\iA(4),B(4),C(4),Z(4),E(8),P(4),R(4),(;1(10) ti=C.87266L1.6E-C!2 E(b)=O. ,I = 3 R(ZI=O. E(lI=O. E(2)=0. E(3)=0. N 2 = 1 N 1 = 1 I=1 DO 150 M=1,720 CALL RUNGI~S(A,R,C,Z,E,t1,J,BETA,G1,P) R(l)=E(2)~:SQRT(BETA/2.0) GOT0(4,5,22I,Nl

4 IF(G1(1)-R(1))7,6,3 3 iF(R(1)-R(2))10,10114C

10 I=1 -1 5 P.! 2 = - 1

pi1=2 IF(Gl(1 )-R(l))20 ,617

20 IF(R(lI-R(2))140121,21 21 I=I+l

48

Table C .1 (Continued)

27 N2=1 N1=3 IF(Gl(I)-R(l))7,6,140

7 F=(G1(1)~(2))/(R(1)~(2)) E(l)=F~:(E(l)-E(5))+E(5) E(2)=F:::(E(2)-E(6))+E(6) E(3)=F:::(E(3)-E(7))+F(7)

6 P(l)=SQRT(2./RETA)+E(3)~:SQRT(EETA/Z.) V=3.1415926F'::Gl(I )+(P(l)Wl(I )-SINIEIl)))

E(4)= E(1):::57.29577957 50 FORMAT(lH2,hHRFTA =,E15.7,6X,3Ht-I =,E15.7///)

PRINT 5O,BETA,H PRINT 100 PRINT 200,E(4),E(2),f(3),Gl(I),P(l),V

200 FORMAT(lH ,OPlF9.3,1P5E16.7) 100 FORElAT(lHO,5X,3HPHI,8X,3HX/B,13X,3HZ/B,13X,3HR/A,13X,3HH/A,

113X,4HV/A3) I=I+N2 IF(I-8)130,130,3

130 IF(I-1)9,140,140 140 R(2)=R(l)

E(5)=E(l) E(6)=E(2) E(7)=E(3)

150 COb!TINUE 9 !'ETURN

E N I>

1,

IMPLICIT Rt-:AL+8 (D),REAL:::4(A-C,E-H,O-Z 1 T>Ir‘;;EI\!SICINA(4),9(4-),C(4),Z(4),E(8),P(4),R(4)

101 FORMAT(lHC!,5X,3HPHI,8X,3HX/B,13X,3HZ/B,13X,3HR/A,l3X,3HH/A, 113%,4HV/k3)

50 FORI~AT(lii1,5HBETA=,E15.7,6X,2HH=,El5.7///) 21?!0 FTjRitlAT(lti ,0PlF9.3,1P5E16.7)

H=0,17453i?E-01 PRIi\!T 50,BETA,H P R I i‘l T !O 1 E (6 1 =c1. J=3 N 2 = i\J 2 / f\i 1 ?.!3=.1746E-01./H r\~ 1 = i\j 3 ::: p; 1 E(1)=0. E(2)=0. E(3)=0. DO 150 i')=1,1\!2

.

-

49

Table C .1 (Continued)

. DO 100 N = l , N l C A L L R UNGKS ( A E3 C 9 Z 9 E ,H J 7 B E T A ( c 7 P 1

100 C O N T I N U E G = E ( 2 ) * S Q K T ( B E T A / 2 . 0 ) E ( 6 ) = E ( 2 1 E ( 7 1 =E ( 3 1 P ( 1 1 = SQR T ( 2 ./ B E T A ) + E ( 7 1 X:SQR T ( BETA /2 . ) V = 3 . 14 159 3:kG * ( P ( 1 1 :;G - S I N ( E ( 1 1 1 1 E ( 4 )=E ( 1 P R I N T

'k57 0295779 57 2 0 0 1 E ( 4 ) ,E ( 6 1, E ( 7 1 ,G , P ( 1 ) 7 V

150 C O N T I N U E R E T U R N END

C C c

C SlJBROUTI NERUNGKD ( D A 9 DB, DC t DZ DE DH 9 J, DB ETA, DG ,DP 1

I F ' I P L I C I T R E A L * 8 (D),REALX:Lt ( A - C t E - H t 0 - Z D I M E N S I O N D A ( 4 ) q D B ( 4 ) t D C ( 4 ) t D Z ( 4 ) t D f ( 8 ) t D P ( 4 ) 7 D R ( 4 ) DA ( 1) =DH:I:OO 16666666667 D A ( 2 ) = D H + 0 . 3 3 3 3 3 3 3 3 3 3 3 [ ) A ( 3 ) = D A ( 2

DR ( 4 ) = o o

DO ( 1 ) =DH'::G05 DB( 2 ) = D R ( 1) D E ( 3 = DH D O Z O I =1, J DC(I ) = O .

20 D Z ( I I = D E ( I 1 D 0 3 0 X = 1 9 4 .

D A ( 4 = m ( 1 I

C A L L RHOD ( D A , DB t D C t DZ ,DE, DH J ,DBETA,DG,DP) D Z ( l ) = l o D O 5 0 1 =1t J D C ( 1 ) = D C ( I ) + D A ( K ) : k D Z (I 1

50 D Z ( 1 ) = D E ( I ) + D B ( K ) : k D Z ( I 1 30 C O N T I N U E

D C ( 1 ) = D H D 0 6 0 1 = 1 , J

6 0 D E ( I ) = D E ( I ) + D C ( I 1 R E T U R N END

C C C

r, SlJBROUT I NE RHOD ( DA 9 D B 7 DC T DZ 7 DE 9 DH 9 J 7 OB ET A 7 DG 9 D P 1

I !$1P L I C I T DI MEFJS I ONDA ( 4 1 9 D B ( 4 1

R E AL +8 ( D I 7 R E A L : k 4 ( A-C 9 E -ti 9 0-Z ) DC ( 4

w T DZ ( 4 1 DE ( 8 7 D P ( 4 1 9 D R ( 4 1

50

Table C .1 (Continued) -

IF(DZ(1))70,75,70 70 DRH=1.0/(2.O+DBETA~:DZ(3)-DSIN(DZ(l))/DZ(2))

GOT076 75 DRH=l.O 76 DZ(2f=DRH:::DCOS(DZ(l))

DZ (3)=DRH+DSIN(DZ ( 1) 1 RETURN END

C C C

SUBROUTINERUNGKS(A,BrC,Z,E,H,J,BETA,G,P)

IMPLICIT REAL’::8(D) ,REAL>::4(A-C,E-H,O-Z) DIMENSIONA(4),B(4),C(4),Z~4),E(B),P(4),R~4~ A( 1 )=H::O.16666666667 A(2)=HG:0.33333333333 A(3)=A(2) A(4)=A(l) B(4)=0. B(l)=Hs0.5 B(2)=B(l) B(3)=H DD2CI=l,J C(1 )=O.

20 Z(1 )=E(I 1 D030!<=1,4 CALLRHOS(A,fi,C,Z,E,H,J,BETA,G,P, Z(l)=l. DO5OI=l,J C(1 )=C(I )+A(K)::‘Z (I 1

50 Z (I )=E(I )+B(K):::Z (I 1 30 CONTINUE

C ( 1) =H DObOI=l,J

60 Et1 )=E(I )+C(I) RETURN END

C C C

SUBROUTINERHOS(A,B,C,Z,E,H,J,BETA,G,P) c

IMPLICIT REAL:::8 (D) ,REAL*4(A-C,E-H,O-Z) DIMENSIONA(4),B(4),C(4),Z(4),E(B),P(4),R(4) IF(Z (1 1 )70,75,70

70 RH=1.0/(2.O+BETA~Z(3)-SIN(Z(1))/Zo)) GOT076

75 RH=l.O 76 Z(2)=RH:kCOS(Z(l))

Z (3)=RH:::SIi’I(Z (1) 1 RETURN END

.

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ORNL-TM-3528

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